H. SOLLER

226

Figure 2. Proposed scheme to produce heralded single photons in arbitrary first-order transverse spatial states: S represents

a Sagnac interferometer. The stripped and stressed 3-mode fiber is pumped with a rotated HG pump laser. The HG laser

pump additionally goes through a nonlinear cry stal.

biphoton wavefunction from now on. In the basis of HG

modes of the two photons it takes the form ([15,16])

2HV VH,

ph nm

(4)

where H and V denote the horizontal and vertical

single-photon polarizations and nm

denotes the spa-

tial part of the biphoton wavefunction. This one we can

further expand in terms of spatial HG modes but since

we limit ourselves to the usage of 3-mode fibers we can

immediately truncate the series and obtain [7]

00 45

HG HGHV VH

nm AB ABAB

. (5)

A and B label the different angular momentum comp-

nents of the wavefunction. Since the Sagnac interfere-

ometer sorts different angular momentum components [7]

these two labels also correspond to the different output

ports of the interferometer.

Now we can use the results gained above. Using the

compression stage in the setup depicted in Figure 2 we

can induce stress and obtain the mode conversion from

45

HG to or as shown in Equations (1)

and (2).

1

0

LG 1

0

LG

Consequently the output photon in A heralds an output

photon in B with a stress dependent well-defined first-

order spatial mode

1

0

LGHVVH.

B

(6)

Indeed these two different compressions possible al-

low to cover the entire Poincaré sphere of first-order

transverse spatial modes [17].

4. Conclusion

In this paper we have generalized the results of [1] as we

have worked with a polarized non-rotated beam and a

true 3-mode fiber that has been stripped in the stress re-

gion. We have investigated different methods of fiber

stripping and have shown that the method works even for

a 3-mode fiber as needed for experimental proposals, e.g.

in [7]. We investigated the heralding of single photons

using our approach.

5. Acknowledgements

The author would like to thank C. C. Leary, M. G. Ray-

mer and B. J. Smith for their hospitality at the University

of Oregon and I. M. Deppner for many interesting dis-

cussions.

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